
Applied Math Applied Math Forum 
 LinkBack  Thread Tools  Display Modes 
August 29th, 2019, 05:53 PM  #11  
Newbie Joined: Aug 2019 From: Canada Posts: 9 Thanks: 0  Quote:
Last edited by skipjack; August 29th, 2019 at 06:03 PM.  
August 29th, 2019, 06:01 PM  #12 
Global Moderator Joined: Dec 2006 Posts: 21,105 Thanks: 2324 
The "inner pentagon" in my diagram is ignored because it doesn't use any lattice points. This wouldn't, as you say, comply with your last revision of the rules, but your motivation for the rules is unclear, as you keep changing them.

August 29th, 2019, 06:19 PM  #13 
Senior Member Joined: Jun 2019 From: USA Posts: 376 Thanks: 202 
My gut instinct is that you're going to have to take connectivity into account for k > 4. I'm guessing your solutions for N3 and N4 bank on counting the number of noncolinear triplets and quadruplets, respectively? If so, the solution for the larger polygons may be fundamentally different. Incidentally, a brute force computer code should easily be able to do this for max(m,n) up to 100s if not 1000s or more. If this were my project, I would probably try a parametric study of simulations first, look for patterns, follow two or three red herrings, and then give up and table it for a few decades. 
August 29th, 2019, 06:21 PM  #14 
Newbie Joined: Aug 2019 From: Canada Posts: 9 Thanks: 0 
That last revision should be the final one  there shouldn't be any other cases left. As long as it forms a simple polygon, it is allowed.

August 29th, 2019, 06:31 PM  #15  
Senior Member Joined: Jun 2019 From: USA Posts: 376 Thanks: 202 
One last clarification, and I think the rules will be clear for everyone: Quote:
In other words, would the triangle {(0,0), (0,2), (2,2)} be an acceptable polygon on a 3x3 grid?  
August 29th, 2019, 06:42 PM  #16  
Newbie Joined: Aug 2019 From: Canada Posts: 9 Thanks: 0  Quote:
 
August 29th, 2019, 06:54 PM  #17 
Newbie Joined: Aug 2019 From: Canada Posts: 9 Thanks: 0 
Solutions that include cases where {(0,0), (0,2), (2,2)} is acceptable are welcome as well, though. It's still better than nothing.


Tags 
combinations, combinatorics, dots, grid, ksided, number, polygons 
Thread Tools  
Display Modes  

Similar Threads  
Thread  Thread Starter  Forum  Replies  Last Post 
Probability of getting a number in a biased six sided dice  simpleman  Probability and Statistics  1  September 3rd, 2016 02:54 PM 
Nine dots  bluehero  Geometry  0  September 13th, 2015 07:13 AM 
Two Rectangles, Two dots, Proportion and Relation  hunter2379  Algebra  2  January 16th, 2013 07:44 AM 
The number of soccer balls to build 4 sided pyramid  simonbinxs  Algebra  5  July 17th, 2012 04:12 PM 
Selfintersecting Polygons and Simple Polygons  telltree  Algebra  2  July 28th, 2009 10:28 AM 